MinMax Arena
中/EN
P83 · Packing and covering · Classic · Hard

Equilateral triangles in a square · n = 33

Pack n unit-side equilateral triangles into a square. Contact is allowed; interiors cannot overlap. Minimise the container square side.

Instancen = 33
ObjectiveMinimize container square side
Best known, unproven4.05164Source-precision interval: 4.05164–4.05165 (values inside match)exhibited site certificate 4.051645123Compiled by Erich Friedman; 33. Found by Jake Loyd in June 2026. Source expression: s = 4.05164+.

Formal definition

  • ContainerThe editor and certificate normalize to the unit square [0,1]². This is a coordinate convention only; scores use the literature's unit-piece scale. Boundary contact is allowed.
  • SubmissionSubmit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4.
  • ConstraintsExactly n congruent shapes lie wholly in the container with pairwise disjoint interiors. Polygon rotations are independent, not limited to quarter turns.
  • PrecisionInput decimals are exact rationals; regular-polygon vertices are algebraic, not rounded templates. Feasibility has no floating-point tolerance. A finite-decimal certificate is not a continuous optimality proof.
  • Objectivecontainer square side = 1/(2 sin(pi/3) r), smaller is better. Pages, leaderboards and literature share these units; conversion is automatic.
Open the full editor ↗
VERIFIED CONSTRUCTIONVERIFIED CONSTRUCTION

Getting a feel for it

Source and certificate

External targets are converted with their printed precision; decimal coordinate certificates are separate. Reconstruction loss never lowers the literature target, and a source construction is not automatically an optimality proof.

Source
The record-holding arrangement for Equilateral triangles in a square n = 33, 4.051645123
Current leader

4.051645123

container square side

Matches the best known
Answer source33. Found by Jake Loyd
Solution methodPublished reference construction
Challenge this record ↗Submit a proof / idea ↓Share a proof or idea in the discussion. Accepted contributions can earn proof points.
Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

ANSWER FORMAT

How to write your answer

the unit square [0,1]²

Submit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4.

The current leader's answer

{
  "placements": [
    {
      "turn": "0.267949192",
      "x": "0.352367660",
      "y": "0.928751129"
    },
    {
      "turn": "0.267949192",
      "x": "0.599180987",
      "y": "0.928751129"
    },
    {
      "turn": "0.267949192",
      "x": "0.845994315",
      "y": "0.928751129"
    },
    {
      "turn": "0.564516019",
      "x": "0.155336674",
      "y": "0.877995633"
    },
    {
      "turn": "-0.267949192",
      "x": "0.475774324",
      "y": "0.857502259"
    },
    {
      "turn": "-0.267949192",
      "x": "0.722587651",
      "y": "0.857502259"
    },
    {
      "turn": "0.000000000",
      "x": "0.071248871",
      "y": "0.759531060"
    },
    {
      "turn": "-0.572971036",
      "x": "0.927940501",
      "y": "0.751700176"
    },
    {
      "turn": "-0.039758310",
      "x": "0.294567982",
      "y": "0.739548160"
    },
    {
      "turn": "0.267949192",
      "x": "0.570603140",
      "y": "0.715004518"
    },
    {
      "turn": "0.577350269",
      "x": "0.142497741",
      "y": "0.636124397"
    },
    {
      "turn": "0.003290665",
      "x": "0.855868611",
      "y": "0.628757938"
    },
    {
      "turn": "0.525528726",
      "x": "0.373770597",
      "y": "0.619428003"
    },
    {
      "turn": "-0.566971486",
      "x": "0.713083458",
      "y": "0.574735085"
    },
    {
      "turn": "-0.165461141",
      "x": "0.520674786",
      "y": "0.521373305"
    },
    {
      "turn": "0.000000000",
      "x": "0.071248871",
      "y": "0.512717733"
    },
    {
      "turn": "-0.032234690",
      "x": "0.292795263",
      "y": "0.499931544"
    },
    {
      "turn": "-0.577350269",
      "x": "0.928751129",
      "y": "0.505358428"
    },
    {
      "turn": "-0.577350269",
      "x": "0.142497741",
      "y": "0.389311069"
    },
    {
      "turn": "-0.000000000",
      "x": "0.857502259",
      "y": "0.381951764"
    },
    {
      "turn": "0.267949192",
      "x": "0.662846724",
      "y": "0.381208794"
    },
    {
      "turn": "0.267949192",
      "x": "0.430446624",
      "y": "0.356244353"
    },
    {
      "turn": "-0.267949192",
      "x": "0.307039960",
      "y": "0.284995482"
    },
    {
      "turn": "-0.267949192",
      "x": "0.553853287",
      "y": "0.284995482"
    },
    {
      "turn": "0.000000000",
      "x": "0.071248871",
      "y": "0.265904405"
    },
    {
      "turn": "0.577350269",
      "x": "0.928751129",
      "y": "0.258545101"
    },
    {
      "turn": "0.267949192",
      "x": "0.246813328",
      "y": "0.142497741"
    },
    {
      "turn": "0.267949192",
      "x": "0.493626655",
      "y": "0.142497741"
    },
    {
      "turn": "0.267949192",
      "x": "0.740439983",
      "y": "0.142497741"
    },
    {
      "turn": "-0.267949192",
      "x": "0.123406664",
      "y": "0.071248871"
    },
    {
      "turn": "-0.267949192",
      "x": "0.370219992",
      "y": "0.071248871"
    },
    {
      "turn": "-0.267949192",
      "x": "0.617033319",
      "y": "0.071248871"
    },
    {
      "turn": "-0.267949192",
      "x": "0.863846647",
      "y": "0.071248871"
    }
  ],
  "radius": "0.142497739"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 33,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.267949192",
      "x": "0.352367660",
      "y": "0.928751129"
    },
    {
      "turn": "0.267949192",
      "x": "0.599180987",
      "y": "0.928751129"
    },
    {
      "turn": "0.267949192",
      "x": "0.845994315",
      "y": "0.928751129"
    },
    {
      "turn": "0.564516019",
      "x": "0.155336674",
      "y": "0.877995633"
    },
    {
      "turn": "-0.267949192",
      "x": "0.475774324",
      "y": "0.857502259"
    },
    {
      "turn": "-0.267949192",
      "x": "0.722587651",
      "y": "0.857502259"
    },
    {
      "turn": "0.000000000",
      "x": "0.071248871",
      "y": "0.759531060"
    },
    {
      "turn": "-0.572971036",
      "x": "0.927940501",
      "y": "0.751700176"
    },
    {
      "turn": "-0.039758310",
      "x": "0.294567982",
      "y": "0.739548160"
    },
    {
      "turn": "0.267949192",
      "x": "0.570603140",
      "y": "0.715004518"
    },
    {
      "turn": "0.577350269",
      "x": "0.142497741",
      "y": "0.636124397"
    },
    {
      "turn": "0.003290665",
      "x": "0.855868611",
      "y": "0.628757938"
    },
    {
      "turn": "0.525528726",
      "x": "0.373770597",
      "y": "0.619428003"
    },
    {
      "turn": "-0.566971486",
      "x": "0.713083458",
      "y": "0.574735085"
    },
    {
      "turn": "-0.165461141",
      "x": "0.520674786",
      "y": "0.521373305"
    },
    {
      "turn": "0.000000000",
      "x": "0.071248871",
      "y": "0.512717733"
    },
    {
      "turn": "-0.032234690",
      "x": "0.292795263",
      "y": "0.499931544"
    },
    {
      "turn": "-0.577350269",
      "x": "0.928751129",
      "y": "0.505358428"
    },
    {
      "turn": "-0.577350269",
      "x": "0.142497741",
      "y": "0.389311069"
    },
    {
      "turn": "-0.000000000",
      "x": "0.857502259",
      "y": "0.381951764"
    },
    {
      "turn": "0.267949192",
      "x": "0.662846724",
      "y": "0.381208794"
    },
    {
      "turn": "0.267949192",
      "x": "0.430446624",
      "y": "0.356244353"
    },
    {
      "turn": "-0.267949192",
      "x": "0.307039960",
      "y": "0.284995482"
    },
    {
      "turn": "-0.267949192",
      "x": "0.553853287",
      "y": "0.284995482"
    },
    {
      "turn": "0.000000000",
      "x": "0.071248871",
      "y": "0.265904405"
    },
    {
      "turn": "0.577350269",
      "x": "0.928751129",
      "y": "0.258545101"
    },
    {
      "turn": "0.267949192",
      "x": "0.246813328",
      "y": "0.142497741"
    },
    {
      "turn": "0.267949192",
      "x": "0.493626655",
      "y": "0.142497741"
    },
    {
      "turn": "0.267949192",
      "x": "0.740439983",
      "y": "0.142497741"
    },
    {
      "turn": "-0.267949192",
      "x": "0.123406664",
      "y": "0.071248871"
    },
    {
      "turn": "-0.267949192",
      "x": "0.370219992",
      "y": "0.071248871"
    },
    {
      "turn": "-0.267949192",
      "x": "0.617033319",
      "y": "0.071248871"
    },
    {
      "turn": "-0.267949192",
      "x": "0.863846647",
      "y": "0.071248871"
    }
  ],
  "radius": "0.142497739"
}

Submit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4. · Verifier v1.0.0

DISCUSSION

Discussion

Talk strategy, share methods, ask why you are stuck. Every signed-in user can post. Posts carry your public byline, the same name your records use; the #number after it is the account's signup ordinal, so a name cannot be worn by someone else. New posts appear after an automated review.

No threads yet. Be the first to talk about this problem?